Problem 13 - Beginning and Intermediate Algebra

Question

Find each sum. $$ -5+(-7) $$

Step-by-Step Solution

Answer

-12

Step 1: Identify the numbers

Look at the numbers in the expression. We have -5 and -7.

Step 2: Apply the rule for adding negative numbers

When adding two negative numbers, the sum is negative. We add the absolute values of the numbers and then put a negative sign before the result.

Step 3: Add the absolute values

The absolute value of -5 is 5 and the absolute value of -7 is 7. Add these absolute values together: 5 + 7 = 12.

Step 4: Apply the negative sign

Since both numbers we are adding are negative, the result should also be negative. Therefore, the sum is -12.

Key Concepts

Integer Addition

Adding integers is a fundamental concept in math. Integers are all the whole numbers and their negatives, like -2, 0, and 3. When adding two integers, there are a few simple rules to follow.

First, remember that the addition of two positive integers results in a positive integer. For example, 2 + 3 = 5.

Similarly, adding two negative integers yields a negative integer. For instance, -5 + (-7) = -12. This happens because each negative integer represents a movement to the left on the number line.

When adding a positive integer and a negative integer, the signs determine the outcome. If the positive integer is larger in absolute value, the result is positive. If the negative integer has a greater absolute value, the result is negative. Here is where the next concept, the absolute value, plays an important role in understanding the signs.

Absolute Value

The absolute value of a number is its distance from zero on the number line, without considering direction.

It is always a positive number. For example, the absolute value of -7 is 7, written as \(|-7| = 7\). Similarly, the absolute value of 5 is just 5, written as \(|5| = 5\).

Understanding absolute values helps simplify problems involving integers of different signs. When you know the absolute values, you can easily determine the result of addition.

For the problem -5 + (-7), you simply add the absolute values: \(|-5| + |-7| = 5 + 7 = 12\). Then, apply the negative sign to the sum since both original numbers were negative resulting in -12.

Negative Numbers

Negative numbers represent quantities less than zero. They are positioned to the left of zero on the number line. Understanding negative numbers is essential for integer operations.

When adding negative numbers:

• Think of moving further left on the number line.
• Each step you take to the left reduces the value more.
• The more negative a number, the further left it is on the number line.

For instance, in the expression -5 + (-7), start at -5 on the number line. Moving 7 steps to the left lands you on -12.

Remember, the sign rules:

• Adding a negative number to another negative number increases the magnitude of the negative result.
• Add the absolute values and keep the negative sign.

Mastering these concepts makes integer addition much easier and helps with more complex math problems later on.